of Edinburgh, Session 1874 - 75 . 
579 
logarithm.” In other words, when we wish to calculate logarithms 
to 14 decimals, making use of 6 orders of differences, the approxi- 
mation being carried for the 1st order to 16, for the 2d to 18, the 
3d to 20, the 4th to 22, the 5th to 24, and the 6th to 26, the error 
resulting from an uncertainty in the 26th figure is multiplied after 
200 terms by 82 472 326 300. 
To see exactly the state of matters, let us make use of algebraic 
signs. Giving to the letters the meaning which I have assigned to 
them in my memoir inserted in the 4th volume of the “ Annales 
de TObservatoire,” we have for the determination of the final 
logarithm u v in terms of the initial logarithm and of the succes- 
sive differences up to the sixth order, 
If we denote by E 0 , E x , E 2 , . . . . the greatest error which 
arises in the calculation of u 0 , Au 0 , A u 0 .... we have, on giving 
the same sign to all the errors, in order to obtain the greatest pos- 
sible error in the result. 
putting each error in the differences at ^ of a unit in its own last 
place, and making p = 200, the successive coefficients, have the fol- 
lowing values— 200 , 19900 , 13 13400 , 64 684 950 , 2 535 650 040 , 
82 408 626 300 . 
Thus, the greatest final error in the value of u m determined 
by the method of differences must be less than 0 - 5 + 1*0 + 0-7 
+ 0-5 + 0'2 + 0'05 = 3*95 in the fourteenth place; so that by the 
simple repetition of the errors made in the elements of the calcula- 
tion, the total error can never rise to more than four units in the 
14th decimal place. We see thus how fantasmagoric is the number 
82 472 326 300 (inaccurate besides) which is given by Mr Sang. 
In point of fact, the error really rises higher than this in the 
Cadastre tables, but that is because of the differences omitted. 
'0 
4 • . .^- S A % 0 + . . . ,t 4 A% 4 . . . . V 6 AX . 
4 5 o 
E* - E 0 + p\ + p ^ E 2 + 
P-2 V ~ 3 V- 4 P~ 5 e 
F 2 3 4 5 6 6 ‘ 
4 F 
VOL. VI! I. 
